Figure on a minimum of six days Virginia to GC or Yellowstone, six days return, three days minimum between Yellowstone and GC. (That's 300+ miles a day by the shortest routes, without doing anything along the way.) That leaves 13 non-driving days.
When I was single and staying in motels, I could expect 500 to 600 miles a day, but that was by making good time in a BMW. Now, pulling a fifth-wheel, even if I wanted to do more than 300 miles day after day, my wife would be impossible to live with. And I don't travel with kids.
You could use all your non-driving days in GC and Yellowstone. Or, you could use them somewhere else. You can work that out. But (for me at least), the planning of a trip is a big part of its enjoyment.
